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Let $k$ be a positive real number. The square with vertices  (k, 0), (0, k), (-k, 0), and (0, -k) is plotted in the coordinate plane.
Find conditions on a > 0 and b > 0 such that the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] is contained inside the square (and tangent to all of its sides). HINTS: Suppose that the line x+y=k is tangent to the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.\]. Algebraically, what can we say about the solutions? In particular, the number of solutions?

 

 May 22, 2020
 #1
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What's an ellipse?

 
 May 22, 2020
 #2
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Let \(k\) be a positive real number. The square with vertices  \((k, 0)\), \((0, k)\), \((-k, 0)\), and \((0, -k)\) is plotted in the coordinate plane.
Find conditions on \(a > 0\) and \(b > 0\) such that the ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) is contained inside the square (and tangent to all of its sides).

 

I Suppose that the line \(\mathbf{y=x+ k}\) is tangent to the ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\).

My answer see here: https://web2.0calc.com/questions/question-about-conics-pls-help#r3

 

laugh

 
 May 22, 2020
 #3
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@heureka, what do $x_t$ and $y_t$ mean in your solution? Are they bases from logs?\(\)

 
Guest May 22, 2020
 #4
avatar+24949 
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@heureka, what do $x_t$ and $y_t$ mean in your solution? Are they bases from logs?

 

no

 

\((x_t,y_t)\) are the coordinates of the tanget point P at the ellipse on my line \(y = x+k\). See my link and the graph.

The tanget point of the circle is \((-10,10)\), so \(x_t=-10\) and \(y_t =10\).

 

laugh

 
heureka  May 22, 2020
 #5
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@heureka, what does $1= -\frac {b^2}{a^2} \frac{x_t}{y_t}$ mean in your solution? Can you explain to me what that means?

 
Guest May 23, 2020

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